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Properties

Label 2-483-161.160-c1-0-1
Degree $2$
Conductor $483$
Sign $-0.999 - 0.0319i$
Analytic cond. $3.85677$
Root an. cond. $1.96386$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.254·2-s + i·3-s − 1.93·4-s + 0.458·5-s + 0.254i·6-s + (−1.07 + 2.41i)7-s − 8-s − 9-s + 0.116·10-s + 1.34i·11-s − 1.93i·12-s − 4.42i·13-s + (−0.272 + 0.614i)14-s + 0.458i·15-s + 3.61·16-s − 7.09·17-s + ⋯
L(s)  = 1  + 0.179·2-s + 0.577i·3-s − 0.967·4-s + 0.204·5-s + 0.103i·6-s + (−0.405 + 0.914i)7-s − 0.353·8-s − 0.333·9-s + 0.0368·10-s + 0.405i·11-s − 0.558i·12-s − 1.22i·13-s + (−0.0728 + 0.164i)14-s + 0.118i·15-s + 0.904·16-s − 1.72·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0319i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.0319i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $-0.999 - 0.0319i$
Analytic conductor: \(3.85677\)
Root analytic conductor: \(1.96386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{483} (160, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 483,\ (\ :1/2),\ -0.999 - 0.0319i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00634792 + 0.397706i\)
\(L(\frac12)\) \(\approx\) \(0.00634792 + 0.397706i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - iT \)
7 \( 1 + (1.07 - 2.41i)T \)
23 \( 1 + (4.44 - 1.80i)T \)
good2 \( 1 - 0.254T + 2T^{2} \)
5 \( 1 - 0.458T + 5T^{2} \)
11 \( 1 - 1.34iT - 11T^{2} \)
13 \( 1 + 4.42iT - 13T^{2} \)
17 \( 1 + 7.09T + 17T^{2} \)
19 \( 1 + 4.40T + 19T^{2} \)
29 \( 1 - 4.18T + 29T^{2} \)
31 \( 1 - 4.31iT - 31T^{2} \)
37 \( 1 - 8.56iT - 37T^{2} \)
41 \( 1 + 3.81iT - 41T^{2} \)
43 \( 1 - 7.18iT - 43T^{2} \)
47 \( 1 + 7.23iT - 47T^{2} \)
53 \( 1 + 8.35iT - 53T^{2} \)
59 \( 1 - 8.10iT - 59T^{2} \)
61 \( 1 - 10.2T + 61T^{2} \)
67 \( 1 - 5.84iT - 67T^{2} \)
71 \( 1 - 8.42T + 71T^{2} \)
73 \( 1 - 7.04iT - 73T^{2} \)
79 \( 1 - 6.86iT - 79T^{2} \)
83 \( 1 - 4.95T + 83T^{2} \)
89 \( 1 + 16.0T + 89T^{2} \)
97 \( 1 - 3.86T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.43790579481275765245660272805, −10.19565650002296032464433359028, −9.780431258960005177429457654610, −8.667173327643504519899283987531, −8.297064124048687422635230287362, −6.55880702194353143423391932953, −5.62937718038922193629530805330, −4.77380767157110667627030218899, −3.75120551174955860277744025883, −2.44939828587195326512387526671, 0.22025012169095997767510772350, 2.14048648471998984580123345456, 3.91622570489640318298763060835, 4.49331695899610248933747097665, 6.05617552874984122088985852976, 6.67412406480129628601280189259, 7.86946817151013008192407902493, 8.809312257551837835544455459273, 9.504516058399405969110407609629, 10.55746291313127049182162651210

Graph of the $Z$-function along the critical line